I was actually thinking of writing up an article comparing the Villages at some point, so I guess I'll just fold those thoughts into a mega-post here.
There is also one other card which gives +2 Actions that nobody's mentioned yet: Nobles! Judged strictly by its Village effect, Nobles is horrible: +2 Actions for $6 is as inefficient as it gets. There are surely some corner cases where buying Nobles for the Actions is actually worth it (perhaps a set with Goons, Scrying Pool, Chapel, and lots of cantrips but no other Villages), but by and large you want to be using Nobles as a points-giving Smithy, preferably with other Villages on the board. So I'll refrain from ranking Nobles.
Anyway, the other Villages, which I've grouped into several tiers and attempted to rank:
TIER 1
1. Fishing Village- The obvious best Village; I think it's safe to say everyone agrees with this one. It's just such a good value! $1 this turn and next makes it almost as good at giving cash as Silver, and then it gives you two extra actions to boot, which is more than any other card in the game! To give you a sense of just how good it is for its price, take a look at Caravan, which is supposed to be one of the best $4s. Caravan is nothing the turn you play it, and a Lab the next turn. Fishing Village is +$1, +Action, -Card this turn and (as Rinkworks pointed out) a Bazaar the next- and it's cheaper! Basically, if you're buying terminals at all (and not going Bank), there's virtually no reason to buy Silver ahead of this, ever. The one supposed downside to FV is the lack of +Card the turn you play it, which it shares with Festival and University. So Library/Watchtower/Menagerie become more powerful, and +Card in general is necessary for chaining. It's better than Festival and University in the absence of those enablers, though, because a) it's cheap and b) the Duration effect mitigates that downside. Another big point in Fishing Village's favor, one which it only shares with the powerful but expensive Bazaar and Festival, is that it gives cash along with actions, which allows you the crucial ability to bake buying power into your engine, and not muck around as much with inert Treasure cards. This is actually really important when setting up many engines, and FV does it for two less than its competition.
2. Hamlet- Hamlet and FV are the two Villages which deserve spots on the Top 5 lists (well, maybe NV does just because there are so few $2s); Hamlet is probably the second-best $2. The key to Hamlet's power is its extreme flexibility and cheapness. It's not great in trimmed decks where discarding is painful, but usually you can dump your Coppers and Estates easily to get the effects you need, and there's virtually never a downside to having more Hamlets. And it works wonders with Library/Watchtower/Menagerie, like many of the Villages here in fact.
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TIER 2 Like Tier 1, Tier 2 cards are excellent engine cards and are worth buying in most situations. But while the Tier 1 cards are cheap enough to basically always be worth buying even when their effects aren't super crucial, the three Tier 2 cards are expensive- they're all $5, and they have to fight against other $5 cards to get bought. So they don't make any Best Of lists. But they're usually good to have around, and their power is hard to ignore.
3 and 4 Bazaar/Festival- I've explained why I like Bazaar so much in the post that inspired this thread.
Bazaar is probably the best Village for when +Card is scarce, and/or when you have sources of +Buy. Festival (which is my favorite card in the game) is its mirror image: it desperately needs good +Card in order to be anything more than a super-Woodcutter, but it is the best Village at giving +Coin and +Buy. Obviously the Library/Watchtower/Menagerie trio is great, but Festivals play particularly well with Minion and Tactician as well. In Festival/Minion games, I love to spend my first $5 on a Festival, to take advantage of the Buy as fast as I can. This will often pay off in double Minion turns down the line, or lots of extra Lighthouses, or whatnot. I think Festivals can work as an engine with +3 Card effects, despite the lack of Card- basically, think of them as Worker's Villages that always draw Silver, not bad. But if there's no +3 Card, or there's lots of other Buy, Bazaar is the right choice.
If pressed, I would probably rank Bazaar a hair above Festival, but really it's too close to call. Bazaar is good in a wider variety of situations, but Festival is more powerful when it's good, and is at least somewhat useful most of the time as well. In games with both Bazaar and Festival, and not with specific Festival-bait cards, I think the right answer is to buy one or two Festivals (for the Buy and cash) and Bazaars the rest of the way (for the Card).
5. City- Unlike Bazaar (always good to have, rarely spectacular), Cities are often quite bad to have, but when they're good they're so good. In games where piles are liable to run out, Cities are incredibly strong- having a stack of Level 2 Cities is about as good a deck full of frickin' Trusty Steeds; in games where the first pile likely to run out is the Provinces, the City trap will just kill you. The best situation for Cities are setups that allow you to gain them while still boosting your buying power (University, Horn of Plenty), or Curse games. Games with spammable stacks are good too. Games with good enablers for fast Big Money strategies (say Vault, Hoard, Masquerade) are good reason to avoid Cities entirely.
I am generally much more wary of using Cities to enable most engines than I am of using Bazaars or even plain Villages. However, in setups where piles are likely to run out with a decent amount of game left, the Cities become engines themselves.
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TIER 3 There isn't really any unifying theme to the Tier 3 cards. Most of them fall into the "sorta like vanilla Village but a little better" category, but there's one exception, a card that's sometimes much better and sometimes much worse.
6. Farming Village- Probably the best Village-plus, IMO, since it's guaranteed to draw a useful card. It's best in situations where your deck has a lot of junk to sift through, and/or when you can get rid of those Coppers. An incredible (and incredibly obvious) counter to Rabble. Definitely one of the better combo-enablers, especially when paired with weaker +Card effects.
7. Mining Village- Mining Village is the best Village to have when you don't need Villages, and one of the worst to have when you do need them. The true power of Mining Village is not that it efficiently enables action chains (it's one of the worst Villages for that), but that you can open with it and trash it quick for a Turn 3 Gold. It'll chain if it has to (and if it does, its endgame potential for "cashing in" can't be ignored), but the reason it's this high is that it's one of the very few Villages worth buying even if you don't want to chain actions.
8. University- This is obviously the one Tier 3 card which isn't just a Village-plus. Rather, it's a Festival variant which is potentially much more powerful but at the expense of being harder to get and somewhat inflexible. Similar upside (great w/Library/Watchtower; get lotsa good cards) and similar downside (lack of +Card can hurt). Worth grabbing a few if there are Minions or Torturers or Cities to rush, but it forces you to open Potion and it's far less useful than Festival once it's time to go green. Much like City (for which it is one of the best enablers), its upside is tremendous but it gets docked a few ranks due to the fact it is often just an expensive distraction.
9. Worker's Village- Is the extra $1 worth just adding a +Buy? Usually. You often will buy a regular Village for $4, and +Buy is important to most engine decks. That's the big reason I rank Festival so highly. But Worker's Village is kinda low-ish because it's actually a fairly poor source of +Buy. Festival gives you guaranteed money to spend on said Buy; WV merely replaces itself and adds an Action and Buy to the queue. (Of course, there are setups where the +Card is better than +2 Coin, but not usually IMO.) It's excellent for enabling Peddlers, Conspirators, and other cheap cantrips, but unlike the Villages above it, it is merely an enabler.
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TIER 4 This is the lowest tier of Villages. I thought about making a Tier 5, but the fact is that none of the Villages we've seen so far are actually bad cards; there are no Villages I'd put on any "worst of" lists, not even Walled Village. But these are the worst of the bunch, as they tend to provide few or uncertain benefits beyond their Village effect. If you don't have a specific engine in mind, these cards are usually skippable.
10. Village- Wheeee. The original! There aren't many situations where plain Village is the best one for the particular engine, but it's cheaper than the more expensive variants and sometimes you don't need their bonuses. Vanilla Village is still an important card, Village Idiots notwithstanding.
11. Native Village- When I first saw NV, I thought it sucked. Then I realized you could buy a bunch and save up for mega-turns, and I started buying NV all the time. But it's fallen in my estimation a lot recently, and now I think it doesn't just "look worse" than Village, but usually it is worse. If you don't have a lot of NVs, the mat becomes risky; if you have to draw it when there's only one or two cards on there, or it makes you stow away one of your good cards before you wanted to draw the mat, the NV is providing serious card *dis*advantage. You need to mass buy them for NV to be effective, and there's a lot of opportunity cost in that.
NV is very strong for certain megaturn strategies, of course. But if you don't have a specific plan for it, don't buy it. There are so few $2 cards that I might actually consider it a top 5 $2 card despite its low rank, but I think I agree with guided that it belongs at #6.
12. Walled Village- Not the worst! But close to it. WV's benefit is usually not worth the extra $1- in particular, it plays very poorly with cantrips; it requires a deck that needs *terminals* to actually be worth it. It's theoretically nice with cards that trash themselves (Island, Embargo, etc.) but I think WV's best use is with Torturer- it lets you buy fewer villages than would usually be necessary to get the first double-Torturer hit, in fact one is sufficient for much longer than would normally be necessary. In general WV is good in decks with a few strong terminals, and weak in longer chains.
13. Shanty Town- Bringing up the rear, Shanty Town is the closest a Village comes to deserving a spot on any Worst list, though I don't think even Shanty Town would quite fit. The problem is that Shanty Town is uniquely horrible at building many engines: if you have Actions, it doesn't give you Card, and if it gives you Card, that means you're unlikely to use both Actions. And it is the worst card to rely on multiples of: extreme case, compare a hand of five Shanty Towns to a hand of five Villages. This is not to say Shanty Town is useless! One or two Shanty Towns can often be a good buy early in an Action-light deck, to use as a pseudo-Lab. And it plays well with non-terminals: clear out the non-terminals, and play Shanty Town to draw two. But it is pretty much the worst Village if you have to buy multiples, and it is also the worst Village at enabling the play of multiple terminals, which is the whole point of villages after all.